Web Space As a basic guide on how to choose a web host you will want figure out how many mega/gigabytes of web space you will need for your files.
You will want to develop a sense of how many visitors you will expect to receive. I would expect a personal website to get less traffic than a business website with a marketing program behind it to promote it.

Bandwidth The measurement of the number of visitors and the total file size of all of the files you expect to be viewed in a month is call the bandwidth.
The total capacity of available space on the web host's hard disk is called the web space, measured in megabytes or gigabytes.

Price The cost is another factor, more features, web space and bandwidth come at a higher cost. It all depends on "how good" of a website you need.

SupportMost good web hosts offer telephone technical support which is important, if a problem arises you can speak to someone about it to resolve the problem.
Some companies offer email support which isn't bad but I like telephone support better. If you are going to use Front Page to author your website that isn't a problem most every web host supports that you want Front Page Extensions.

Uptime Reliability is important, modern hardware is pretty reliable so this shouldn't be too much of a problem. Most web hosts offer 99.9% uptime which is an industry standard.

Type of Hosting You can obtain shared hosting which has maybe hundreds of sites on a single web server. Then there is VPS or virtual private server which is like your own server except there may be 10 VPS accounts on a single server.
Then there is dedicated hosting which is only you on a single server. Shared is the cheapest cost and dedicated is the most expensive, VPS is in the middle.

Hardware Just like a regular laptop or desktop the number of processors, processor type and speed also the amount of memory available to you are important, the more the merrier.

Operating System You can also choose which operating system you want to have. You DO NOT need to choose a Windows server because your laptop or desktop has Windows that has nothing to do with it.
You need to know what kind of web software you want to use and what OS it runs on. Windows servers usually cost more, Microsoft needs its cut. Linux is free and well supported/updated and basically runs the internet.